Vector Field Visualization and Data Recovery

Vector Field Visualization and Data Recovery

  1. ggvfields
  2. Data Recovery
  3. Conclusion

ggvfields

  1. Introduction of Problem
  2. Current Solutions
  3. Proposed Solutions ggvfields

Before we define the problem

What is a vector field?

First, what is a vector?

A vector is a mathematical object that has both magnitude and direction.

In \(\mathbb{R}^2\), a vector is represented as an ordered pair of numbers \((x, y)\), where \(x\) and \(y\) are the components of the vector along the \(x\)-axis and \(y\)-axis.

Vectors are arrows pointing from one point to another in the plane.

Before we define the problem

What is a vector field?

A vector field in \(\mathbb{R}^2\) is a function that maps each point \((x, y)\) to a vector in the plane.
If \(\mathbf{V}\) is a vector field in \(\mathbb{R}^2\), then:

\(\mathbf{V}: \mathbb{R}^2 \rightarrow \mathbb{R}^2\)

where \(\mathbf{V}(x, y) = (u(x, y), v(x, y))\) assigns to each point \((x, y)\) a vector \((u(x, y), v(x, y))\).

Before we define the problem

There are many applications across various fields:

  1. Physics:
    • Representing force fields (gravitational, electric, magnetic)
    • Describing the velocity field of particles in a fluid flow
  2. Engineering:
    • Modeling airflow over wings and around vehicles
  3. Meteorology:
    • Mapping wind velocities or ocean currents

How do we visualize a vector field?

The Problem: R

library(tidyverse)

x <- seq(-10, 10, by = 1)
y <- seq(-10, 10, by = 1)

grid <-
  expand_grid(x = x, y = y) |> 
  mutate(u = -y, v = x) |> 
  mutate(norm = sqrt(x^2 + y^2)) |> 
  mutate(u = u/max(norm), v = v/max(norm))

ggplot(grid, aes(x = x, y = y)) +
  geom_segment(
    aes(xend = x + u, yend = y + v), 
    arrow = arrow(length = unit(0.1, "cm"))
    ) +
  theme_minimal()

The Problem: R

The Problem: R

  • This is a lot of syntax
  • This requires a large burden on the user
  • It takes a lot to customize the plot
  • It is not even that attractive

Current Solutions: Python

import numpy as np
import matplotlib.pyplot as plt

# Create a grid of points
x = np.linspace(-10, 10, 20)
y = np.linspace(-10, 10, 20)
X, Y = np.meshgrid(x, y)

# Define the vector field
U = -Y
V = X

# Plot the vector field
plt.quiver(X, Y, U, V)

Current Solutions: Python

Current Solutions: Mathematica

VectorPlot[{x, -y}, {x, -3, 3}, {y, -3, 3}]
# VectorPlot[{{x, -y}, {x^2, y^2}}, {x, -3, 3}, {y, -3, 3}]

Proposed Solutions: ggvfields

Requirements

  • Correct
  • Attractive
  • Simple
  • Familiar

Proposed Solutions: ggvfields

Install from Github

remotes::install_github("dusty-turner/ggvfields")

Load the package

library(ggvfields)

geom_vector_field()

Let’s explore the vector field defined by the function:

\[\mathbf{V}(x,y) = (-y, x)\]

f <- function(v) {
  x <- v[1]
  y <- v[2]
  c(-y, x)
}

A very simple and familiar implementation

ggplot() +
  geom_vector_field(fun = f, xlim = c(-10, 10), ylim = c(-10, 10))

geom_vector_field()

ggplot() +
  geom_vector_field(fun = f, xlim = c(-10, 10), ylim = c(-10, 10))

geom_vector_field()

By default, center and scale by the norm \(\mathbf{w} = (u, v)\) given by:

\[|\mathbf{w}| = \sqrt{u^2 + v^2}\]

ggplot() +
  geom_vector_field(fun = f, xlim = c(-10, 10), ylim = c(-10, 10), 
                    center = TRUE, normalize = TRUE
                    )

geom_vector_field()

If desired, we could also map the norm to a number of aesthetics. For example

Color:

ggplot() +
  geom_vector_field(
    aes(color = after_stat(norm)),
    xlim = c(-10, 10), ylim = c(-10, 10), 
    fun = f 
    )

Or alpha:

ggplot() +
  geom_vector_field(
    aes(alpha = after_stat(norm)),
    xlim = c(-10, 10), ylim = c(-10, 10), 
    fun = f 
    )

Calculus Concepts in Vector Fields

\(\mathbf{F} = \langle \mathbf{F}_x(x,y), \mathbf{F}_y(x,y) \rangle\)

Divergence

The divergence of a vector field \(\mathbf{F}\) in \(\mathbb{R}^2\) is defined by: \[\text{div} \, \mathbf{F} = \frac{\partial \mathbf{F}_x}{\partial x} + \frac{\partial \mathbf{F}_y}{\partial y}\] Indicates how much the vector field spreads out or converges at a point.

Curl

The curl of a vector field \(\mathbf{F}\) in \(\mathbb{R}^2\) is defined by: \[\text{curl} \, \mathbf{F} = \frac{\partial \mathbf{F}_y}{\partial x} - \frac{\partial \mathbf{F}_x}{\partial y}\] Measures the rotation or swirling strength at a point.

Laplace Operator

The Laplacian of the vector field \(\mathbf{F}\) is given by: \[\Delta \mathbf{F} = \frac{\partial^2 \mathbf{F}_x}{\partial x^2} + \frac{\partial^2 \mathbf{F}_x}{\partial y^2} + \frac{\partial^2 \mathbf{F}_y}{\partial x^2} + \frac{\partial^2 \mathbf{F}_y}{\partial y^2}\] Describes how the vector field spreads out or compresses at different points.

Directional Derivative

The directional derivative of a vector field measures the rate of change of the field in a specified direction. It is given by: \[D_{\mathbf{v}} \mathbf{F} = \frac{\partial \mathbf{F}_x}{\partial x} v_x + \frac{\partial \mathbf{F}_x}{\partial y} v_y + \frac{\partial \mathbf{F}_y}{\partial x} v_x + \frac{\partial \mathbf{F}_y}{\partial y} v_y\] Represents the rate of change of each component of the field in a given direction.

geom_vector_field()

ggplot() +
  geom_vector_field(
    aes(color = after_stat(divergence)),
    xlim = c(-10, 10), ylim = c(-10, 10), 
    fun = f 
    )

geom_vector_field()

ggplot() +
  geom_vector_field(
    aes(color = after_stat(curl)),
    xlim = c(-10, 10), ylim = c(-10, 10), 
    fun = f 
    )

geom_vector_field()

g <- function(v) {
  x <- v[1]
  y <- v[2]
  c(-sin(y), cos(x))
}

ggplot() +
  geom_vector_field(
    aes(color = after_stat(laplacian)),
    xlim = c(-10, 10), ylim = c(-10, 10), 
    fun = g 
    )

geom_vector_field()

  • future work

geom_streamplot

  • show base code to do this
  • show a lot of reasonable variations
  • future work

geom_complex_function

  • show base code to do this
  • show a lot of reasonable variations
  • future work

  • should i show inner code?
  • do i need to show how ggplot2 layers work?
  • how much variation do i need to do on all the options?
  • should i show how each of these functions work in other languages

Data Recovery

  1. Introduction of Problem
  2. Current Solutions
  3. Proposed Solution

The Problem:

Imagine one of several scenarios
- Researcher who desires to replicate another’s research
- Researcher who wants to use an previous study’s results as a prior for their own

Current Solutions

By hand

  • slow
  • error prone
mtcars |> 
  ggplot(aes(hp, mpg)) +
  geom_point()

Current Solutions

  • email authors
  • many reasons why this could go wrong
  • such as unwilling/unable to share data

Current Solutions

plotdigitizer
automeris.io
engegue-digitizer
(show images of this)

  • Problems: require user interaction, technical expertise, not 100% accurate

ChatGPT

  • Problems: cost money, not 100% accurate, different answers every time asked

Proposed Solution

  1. Neural Network Solution

  2. Methodology

    • All Assumptions made in first model
    • Data Generation
    • AWS implementation (details)

  3. Challenges

    • Money
    • Time
    • Size of neural network

  4. Preliminary results with assumptions

  5. Preliminary results by removing assumptions

  6. Current unsolved issues

    • Unlimited y axis
    • Learn y axis?

  7. Where to go from here

Conclusion Slides

MAJ Dusty Turner